*3.1. The Finite-Di*ff*erence Method*

In order to transform Equations (13) and (14) to first order equations, new independent unknowns will be defined as follows:

*<sup>w</sup>*(ξ, η), *<sup>z</sup>*(ξ, η), *p*(ξ, η), and *g*(ξ, η), where the temperature variable <sup>θ</sup>(ξ, η) is replaced by *g*(ξ, η), and

$$\begin{aligned} F &= \; w, \\ w' &= \; z, \\ \mathbf{g}' &= \mathbf{p}, \end{aligned} \tag{22}$$

Thus, the Equations (13)–(15) are converted to:

$$\begin{cases} \frac{\rho\_f}{\rho\_{nf}} \frac{1}{(1-\chi)^2} \Big(1+\frac{1}{\beta}\Big) \mathbf{z}' + (1+\xi\cot\xi)F\mathbf{z} - w^2 - \frac{\rho\_f\alpha\_{nf}}{\rho\_{nf}\sigma\_f} \mathsf{M}w\\ + \left(\frac{\chi\rho\_s\beta\_s\gamma\beta\_f + (1-\chi)\rho\_f}{\rho\_{nf}}\right) \log\frac{\sin\xi}{\xi} + \frac{\varrho}{4} \frac{\sin\xi\cos\xi}{\xi} = \xi \Big(w\frac{\partial w}{\partial \xi} - z\frac{\partial F}{\partial \xi}\Big), \end{cases} \tag{23}$$

$$\frac{1}{\Pr} \left( \frac{k\_{nf}/k\_f}{(1-\chi) + \chi \{\rho c\_p\}\_s/\left(\rho c\_p\right)\_f} \right) p' + (1 + \xi \cot \xi) F p = \xi \left( w \frac{\partial g}{\partial \xi} - p \frac{\partial F}{\partial \xi} \right) \tag{24}$$

Subject to:

$$w(\xi,0) = F(\xi,0) = 0, \ g(\xi,0) = 1,$$

$$w(\xi,\infty) = \frac{3}{2} \frac{\sin \xi}{\xi}, \ g(\xi,\infty) = 0,\tag{25}$$

where the prime notation denotes the 1st derivative with respect to η,

Next the finite-difference form of Equation (22) for the midpoint (ξ*<sup>n</sup>*, <sup>η</sup>*j*−1/2) of the segment, and find the finite difference form of Equations (23) and (24) about the midpoint (ξ*<sup>n</sup>*−1/2, <sup>η</sup>*j*−1/2) of the rectangle have been obtained as:

$$\left(F^n\_j - F^n\_{j-1} - \frac{h\_j}{2} \{w^n\_j + w^n\_{j-1}\} = 0.\right) = 0. \tag{26}$$

$$w\_j^n - w\_{j-1}^n - \frac{h\_j}{2} \left( z\_j^n + z\_{j-1}^n \right) = 0. \tag{27}$$

$$\mathbf{g}\_{\dot{j}}^{n} - \mathbf{g}\_{\dot{j}-1}^{n} - \frac{h\_{\dot{j}}}{2} (p\_{\dot{j}}^{n} + p\_{\dot{j}-1}^{n}) = 0.\tag{28}$$

ρ*f* ρ*n f* 1 (<sup>1</sup>−<sup>χ</sup>)2.5 -1 + 1β *znj* − *znj*−1 + - *A*+α4 *hj*(*Fnj* + *Fnj*−<sup>1</sup>)(*znj* + *<sup>z</sup>nj*−<sup>1</sup>) − - 1+α4 *hj*(*wnj* + *<sup>w</sup>nj*−<sup>1</sup>)<sup>2</sup> +- α2 *hjzn*−<sup>1</sup> *<sup>j</sup>*−1/2(*Fnj* + *Fnj*−<sup>1</sup>) + 12 χρ*s*(β*s*/β*f*)+(<sup>1</sup>−<sup>χ</sup>)ρ*<sup>f</sup>* ρ*n f* sin *xn*−1*l*<sup>2</sup> *xn*−1*l*<sup>2</sup> <sup>λ</sup>*hj*(*gnj* + *<sup>g</sup>nj*−<sup>1</sup> ) 1ρ*f*δ*nf* αsin*xn*−1*l*<sup>2</sup> cos*xn*−1*l*<sup>2</sup> (29)

− 2  ρ*n f* δ*f Mhj*(*wnj* + *<sup>w</sup>nj*−<sup>1</sup>) − -2 *hjFn*−<sup>1</sup> *<sup>j</sup>*−1/2(*znj* + *<sup>z</sup>nj*−<sup>1</sup>) + 94 *xn*−1*l*<sup>2</sup> *hj* = (*<sup>R</sup>*1)*<sup>n</sup>*−<sup>1</sup> *j*−1/2 1 Pr *kn f* /*k f* (<sup>1</sup>−<sup>χ</sup>)(ρ*Cp*)*f* <sup>+</sup>χ(ρ*cp*)*s*/(ρ*cp*)*fpnj* − *<sup>p</sup>nj*−<sup>1</sup> − α4 *hj*(*wnj* + *<sup>w</sup>nj*−<sup>1</sup>)(*gnj* + *<sup>g</sup>nj*−<sup>1</sup>) <sup>+</sup>*A*+α4 *hj*(*Fnj* + *Fnj*−<sup>1</sup>)(*pnj* + *<sup>p</sup>nj*−<sup>1</sup>) + α2 *hj*(*wnj* + *<sup>w</sup>nj*−<sup>1</sup>)*g<sup>n</sup>*−<sup>1</sup> *j*−1/2 − α2 *hjwn*−<sup>1</sup> *<sup>j</sup>*−1/2(*gnj* + *<sup>g</sup>nj*−<sup>1</sup>) − α 2 *hj*(*pnj* − *<sup>p</sup>nj*−<sup>1</sup>)*Fn*−<sup>1</sup> *j*−1/2 + α2 *hjpn*−<sup>1</sup> *<sup>j</sup>*−1/2(*Fnj* + *Fnj*−<sup>1</sup>) = (*<sup>R</sup>*2)*<sup>n</sup>*−<sup>1</sup> *j*−1/2 (30)

where

$$\alpha = \frac{\mathbf{x}^{n-1l2}}{k\_n}, \ A = \left(1 + \mathbf{x}^{n-1l2} \cot \mathbf{x}^{n-1l2}\right), \ k\_n \text{ is } \Delta\xi, \ \text{and } h\_j \text{ is } \Delta\eta$$

$$(R\_1)^{
\mathbf{n}-1}\_{j-1/2} = -\mathbf{h}\_j \begin{pmatrix} \frac{\rho\_f}{\rho\_{nf}} \frac{1}{(1-\chi)^{2\lambda}} \left(1 + \frac{1}{\beta}\right) \frac{\left(\overline{z}\_j^n - \overline{z}\_{j-1}^n\right)}{h\_j} + (A-\alpha)F^n\_{j-1/2} \mathbf{z}\_{j-1/2}^n \\\ + (\alpha-1) \left(\mathbf{w}\_{j-1/2}^n\right)^2 - \frac{\rho\_f \alpha\_f}{\rho\_{nf} \rho\_f} \Lambda \mathbf{w}\_{j-1/2}^n + \frac{9}{4} \frac{\sin^{n-12} \cos \mathbf{x}^{n-12}}{\mathbf{x}^{n-12}} \\\ + \left(\frac{\chi \rho\_s (\beta\_s/\rho\_f) + (1-\chi)\rho\_f}{\rho\_{nf}}\right) \frac{\sin \mathbf{x}^{n-12}}{\mathbf{x}^{n-12}} \lambda \mathbf{s}\_{j-1/2}^n \end{pmatrix}^{n-1}$$

$$(R\_2)^{\mathbf{n}-1}\_{j-1/2} = -\mathbf{h}\_j \left(\begin{array}{c} \frac{\mathbf{z}\_{\text{n}f}/k\_f}{\Gamma t} \frac{(\mathbf{z}\_f - \mu\_f)}{((1-\chi)/\rho\_{\text{c}f}) + (\chi \rho\_{\text{c}f})\_{\text{s}}/(\mu\_{\text{c}f})}{\rho\_f} \end{array}\right)^{n-1} \tag{31}$$

when ξ = ξ*n* the boundary conditions become:

$$F\_0^{\eta} = \ \ w\_0^{\eta} = 0, \ g\_0^{\eta} = 1,$$

$$\begin{split} w\_l^{\eta} &= \frac{3}{2} \frac{\sin \underline{\xi}}{\underline{\xi}}, \ g\_l^{\eta} = 0, \\ \end{split} \tag{32}$$
